Abstract
We point out the shape characteristics — monotonicity and concavity — of the value functions of optimal economic growth problems. We introduce the concept of shape preservation in approximating the value functions. We also present a shape-preserving algorithm to compute the solutions of continuous-state optimal economic growth problems. Numerical results show that shape-preserving interpolation methods are superior to others with less-sophisticated interpolation in the sense of smaller approximation errors. Scope and Purpose Under standard conditions on a risk-averse utility, the value function of an optimal economic growth problem is known to possess the shape characteristics-monotonicity and concavity. As the closed form solutions are rarely available, the only way to solve for the value function is numerically. However, there are no numerical methods which guarantee to preserve the shape features in the course of approximation. In this article, we introduce the usage of shape preservation and present a shape-preserving interpolation in numerical dynamic programming.
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